Simplify and expand the following expression: $ \dfrac{2x}{x - 2}+\dfrac{2x - 7}{x - 3} $
In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(x - 2)(x - 3)$ Multiply the first term by $\dfrac{x - 3}{x - 3}$ $ \begin{align*} \dfrac{2x}{x - 2} \times \dfrac{x - 3}{x - 3} & = \dfrac{(2x)(x - 3)}{(x - 2)(x - 3)} \\ & = \dfrac{2x^2 - 6x}{(x - 2)(x - 3)}\end{align*} $ Multiply the second term by $\dfrac{x - 2}{x - 2}$ $ \begin{align*} \dfrac{2x - 7}{x - 3} \times \dfrac{x - 2}{x - 2} & = \dfrac{(2x - 7)(x - 2)}{(x - 3)(x - 2)} \\ & = \dfrac{2x^2 - 11x + 14}{(x - 3)(x - 2)}\end{align*} $ Now we have: $ = \dfrac{2x^2 - 6x}{(x - 2)(x - 3)} + \dfrac{2x^2 - 11x + 14}{(x - 3)(x - 2)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{2x^2 - 6x + 2x^2 - 11x + 14}{(x - 2)(x - 3)} $ $ = \dfrac{4x^2 - 17x + 14}{(x - 2)(x - 3)}$ Expand the denominator: $ = \dfrac{4x^2 - 17x + 14}{x^2 - 5x + 6}$